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Chapter 1: Problem 23

Find an equation of the given line. Parallel to \(y=3 x+7 ; x\) -intercept is 2

Short Answer

Expert verified
The equation is \(y = 3x - 6\).

Step by step solution

01

Identify the slope

The slope of the given line, which is parallel to the desired line, is the same as the slope of the equation given. The provided equation is in the form of \(y = mx + b\), where \(m\) is the slope. Here, the slope \(m\) is 3.
02

Find the y-intercept

Given the x-intercept of the desired line is 2, we can determine the y-intercept by setting \(y = 0\) in the line equation and solving for \(b\). Note that at the x-intercept, \(y = 0\). Substituting \(x = 2\) and \(y = 0\) into the line equation: \(0 = 3(2) + b\). Simplifying, we get \(b = -6\).
03

Write the equation of the line

We now know the slope \(m = 3\) and the y-intercept \(b = -6\). Plugging these into the slope-intercept form \(y = mx + b\), we get the equation of the line: \(y = 3x - 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a line is a way to write linear equations. It鈥檚 expressed as \(y = mx + b\), where
  • \(m\) represents the slope of the line
  • \(b\) is the y-intercept, or where the line crosses the y-axis
Understanding this form makes it easy to identify the slope and the y-intercept.For example, in the equation \(y = 3x + 7\), the slope \(m\) is 3, meaning for every unit increase in \(x\), \(y\) increases by 3.The y-intercept \(b\) is 7, meaning the line crosses the y-axis at (0,7).We use this form to easily draft or graph a line and understand its behavior.
Parallel Lines
Parallel lines are lines in the same plane that never intersect. They always have the same slope but different y-intercepts.Given our example, a line parallel to \(y = 3x + 7\) will have the same slope, \(m = 3\).This ensures the lines rise and run at the same rate, never touching each other.To find the exact equation of a parallel line, we use the slope and another point given or derived from context, such as the x-intercept.
X-Intercept
The x-intercept is the point where a line crosses the x-axis. This occurs when \(y = 0\).In our exercise, we are given an x-intercept of 2. This means the line crosses the x-axis at (2,0).We use this information to find the y-intercept of a parallel line by substituting \(x = 2\) and \(y = 0\) into the slope-intercept equation and solving for \(b\).For instance, substituting these into \(y = 3x + b\), we get:\[0 = 3(2) + b\]This simplifies to \(b = -6\).Hence, the equation of the line with an x-intercept of 2 and parallel to \(y = 3x + 7\) is \(y = 3x - 6\).

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Most popular questions from this chapter

Rates of Change Suppose that \(f(x)=-6 / x .\) (a) What is the average rate of change of \(f(x)\) over each of the intervals 1 to 2,1 to \(1.5\), and 1 to \(1.2 ?\) (b) What is the (instantaneous) rate of change of \(f(x)\) when \(x=1 ?\)

Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=\left\\{\begin{array}{ll}2 x-1 & \text { for } 0 \leq x \leq 1 \\ 1 & \text { for } 1

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=x+\frac{1}{x}\)

If \(f(100)=5000\) and \(f^{\prime}(100)=10\), estimate each of the following. (a) \(f(101)\) (b) \(f(100.5)\) (c) \(f(99)\) (d) \(f(98)\) (e) \(f(99.75)\)

Use the limit definition of the derivative to show that if \(f(x)=m x+b\), then \(f^{\prime}(x)=m .\)
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